The fractions 2/3, 5/8, 8/13, 11/18, ... follow a fixed pattern. One later term of the pattern is X/253 and another is 1499/Y. Find the greatest common divisor of X and Y.

Step-by-step solution

1. 1

   Treat the numerators and denominators as two separate sequences. They do not interact, so each can be described on its own. Numerators $2,5,8,11,\dots$ rise by $3$; denominators $3,8,13,18,\dots$ rise by $5$. Both are arithmetic.

2. 2

   Write a formula for the $n$-th term of each. Using $\text{first}+d(n-1)$:

   $\text{numerator}=2+3(n-1)=3n-1,\qquad \text{denominator}=3+5(n-1)=5n-2$

   Spot-check at $n=4$: $3(4)-1=11$ and $5(4)-2=18$, matching $\tfrac{11}{18}$.

3. 3

   Locate the term whose denominator is 253. Solving $5n-2=253$ gives $n=51$ — an integer, so this term really does belong to the pattern. Its numerator is

   $X=3(51)-1=152$

4. 4

   Locate the term whose numerator is 1499. Solving $3n-1=1499$ gives $n=500$, again an integer. Its denominator is

   $Y=5(500)-2=2498$

5. 5

   Factor both numbers before taking the GCD. Guessing is unsafe with a four-digit number:

   $152=2^3\times 19,\qquad 2498=2\times 1249$

   and $1249$ is prime (it is divisible by no prime up to $35$).

6. 6

   Take the common part. The only shared prime is $2$, appearing once in $2498$, so

   $\gcd(152,\,2498)=2$

   The Euclidean algorithm agrees: $2498=16\cdot152+66$, $152=2\cdot66+20$, $66=3\cdot20+6$, $20=3\cdot6+2$, $6=3\cdot2+0$.